Solving Absolute Value Inequalities A Step-by-Step Guide To |2x-4| ≤ 8

Absolute value inequalities can seem daunting at first, but with a systematic approach, they become manageable. This article provides a comprehensive guide to solving the inequality $|2x - 4| \leq 8$, explaining the underlying concepts and step-by-step solution. Understanding absolute value inequalities is crucial not only in mathematics but also in various fields such as physics, engineering, and computer science. Let's dive into how to tackle this problem effectively.

Understanding Absolute Value

Before we delve into the specifics of the inequality $|2x - 4| \leq 8$, it's essential to grasp the concept of absolute value. The absolute value of a number is its distance from zero on the number line. Mathematically, the absolute value of a number $x$, denoted as $|x|$, is defined as:

$|x| = egin{cases} x, & ext{if } x ext{ is greater than or equal to } 0 \ -x, & ext{if } x ext{ is less than } 0

\end{cases}$

In simpler terms, if a number is positive or zero, its absolute value is the number itself. If a number is negative, its absolute value is the positive version of that number. For example, $|5| = 5$ and $|-5| = 5$. This concept is vital because it transforms any number into its non-negative equivalent, focusing solely on its magnitude without considering its direction or sign. When dealing with absolute value inequalities, this means we must consider two cases: one where the expression inside the absolute value is positive or zero, and another where it is negative. This approach ensures that we account for all possible solutions. Recognizing that absolute value represents distance helps visualize solutions on a number line, making the solutions to inequalities more intuitive. The distance interpretation also clarifies why we split absolute value problems into two separate cases, each reflecting a possible direction away from zero. Ultimately, a solid understanding of absolute value is the bedrock for tackling more complex inequalities and equations involving this concept.

Breaking Down the Inequality $|2x - 4|

\leq 8$

To solve the inequality $|2x - 4| \leq 8$, we need to apply the principles of absolute value we discussed earlier. The absolute value inequality $|2x - 4| \leq 8$ essentially states that the distance between $2x - 4$ and 0 must be less than or equal to 8. This condition leads us to consider two separate cases, which cover all possible values of $x$ that satisfy the inequality. The key to understanding how to solve $|2x-4| ≤ 8$ is to break it down into these cases, each representing a different scenario based on the sign of the expression inside the absolute value. By addressing each case individually, we can transform the absolute value inequality into standard linear inequalities, which are easier to solve. The first case assumes that the expression $2x - 4$ is non-negative, and the second case considers when it is negative. This division is critical because the absolute value function behaves differently depending on the sign of its argument. By handling each case meticulously, we ensure that no solution is overlooked, and we accurately identify the range of $x$ values that satisfy the original absolute value inequality. The importance of this method lies in its ability to systematically remove the absolute value, allowing us to apply familiar algebraic techniques to find the solution set.

Case 1: $2x - 4

\geq 0$

In the first case, we assume that the expression inside the absolute value, $2x - 4$, is greater than or equal to zero. This means that $|2x - 4|$ is simply $2x - 4$ itself. Therefore, the inequality becomes:

$2x - 4 \leq 8$

To solve this linear inequality, we add 4 to both sides:

$2x \leq 12$

Then, we divide both sides by 2:

$x \leq 6$

However, we must also consider the initial condition that $2x - 4 \geq 0$. Solving this inequality:

$2x - 4 \geq 0$

Add 4 to both sides:

$2x \geq 4$

Divide both sides by 2:

$x \geq 2$

Combining these results, we find that in this case, the solution must satisfy both $x \leq 6$ and $x \geq 2$. This gives us a solution interval of $2 \leq x \leq 6$. This step is critical because it establishes one part of the overall solution set. Understanding that solving inequalities involving absolute values often requires considering overlapping conditions helps refine the final answer. The intersection of these two inequalities is a bounded interval, showing that the solutions are constrained within specific limits. Thus, the first case not only provides a segment of the solution but also underscores the importance of considering initial conditions when breaking down absolute value problems.

Case 2: $2x - 4 < 0

In the second case, we consider when the expression inside the absolute value, $2x - 4$, is less than zero. In this scenario, $|2x - 4|$ becomes $-(2x - 4)$. Thus, the inequality transforms to:

$-(2x - 4) \leq 8$

Distributing the negative sign, we get:

$-2x + 4 \leq 8$

Subtract 4 from both sides:

$-2x \leq 4$

Divide both sides by -2 (and remember to flip the inequality sign since we're dividing by a negative number):

$x \geq -2$

We also need to consider the initial condition $2x - 4 < 0$. Solving this inequality:

$2x - 4 < 0$

Add 4 to both sides:

$2x < 4$

Divide both sides by 2:

$x < 2$

Combining these results, we find that in this case, the solution must satisfy both $x \geq -2$ and $x < 2$. This gives us a solution interval of $-2 \leq x < 2$. This case is crucial for capturing the complete set of solutions, as it addresses the scenario where the expression inside the absolute value is negative. The process of solving absolute value inequalities requires careful attention to how negative signs affect inequalities, as demonstrated when dividing by -2 and flipping the inequality sign. The intersection of the inequalities $x \geq -2$ and $x < 2$ forms another bounded interval, complementing the solution from the first case. By including this case, we ensure a thorough and accurate solution, demonstrating the comprehensive approach needed for solving absolute value problems.

Combining the Solutions

Now that we have the solutions from both cases, we need to combine them to find the overall solution to the inequality $|2x - 4| \leq 8$. In Case 1, we found that $2 \leq x \leq 6$. In Case 2, we found that $-2 \leq x < 2$. To get the complete solution set, we take the union of these two intervals. This means we include all values of $x$ that satisfy either inequality. The final solution encompasses all $x$ that satisfy $-2 \leq x < 2$ or $2 \leq x \leq 6$. Combining these intervals, we get a single continuous interval: $-2 \leq x \leq 6$. This means that any value of $x$ within this interval will satisfy the original inequality. Visualizing these intervals on a number line can help clarify how they merge to form the complete solution. Combining the individual case solutions is a critical step in how to solve absolute value problems because it ensures that no valid solutions are omitted. The union of the intervals reflects the inclusive nature of the "or" condition, capturing all $x$ values that make the absolute value inequality true. This final step completes the puzzle, providing a clear and concise solution set that accurately represents the answer to the given problem.

Final Solution and Answer

After analyzing both cases and combining the results, we have determined that the solution to the inequality $|2x - 4| \leq 8$ is $-2 \leq x \leq 6$. This means that $x$ must be greater than or equal to -2 and less than or equal to 6. This solution can be expressed in interval notation as $[-2, 6]$. To summarize, the answer to the inequality is the range of values for $x$ that fall within this interval. This range represents all real numbers that, when substituted into the original inequality, make the statement true. The process of arriving at this solution involved breaking down the absolute value inequality into two separate cases, solving each case independently, and then combining the resulting intervals. This systematic approach ensures accuracy and completeness in solving absolute value problems. Understanding that the absolute value represents distance on the number line helps in visualizing and interpreting the solution set. The final interval $[-2, 6]$ provides a clear and concise answer, encapsulating the entire set of valid solutions for the given inequality. Therefore, the correct answer among the provided options is:

D. $x \geq -2$ and $x \leq 6$

This comprehensive solution not only answers the question but also reinforces the method for solving similar absolute value inequalities, making it a valuable guide for anyone studying algebra or related topics.